This expression is equivalent to that given in Refs. [1]} and [2]}.
Based on such expression of heat current, the phonon Hall
conductivity can be obtained through the Green-Kubo formula
[3]}:
\(\kappa _{xy} = \frac{V}{\hbar T} \int _0^{\hbar /(k_B T)}\!\!\!\! d\lambda \int _0^\infty \! dt\, \bigl \langle J^x(-i\lambda ) J^y(t) \bigr \rangle _{\rm eq},\) 
